2024 AMC 12B Problems/Problem 24

Problem 24

What is the number of ordered triples of positive integers, with , such that there exists a (non-degenerate) triangle with an integer inradius for which , , and are the lengths of the altitudes from to , to , and to , respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)

Solution 1

First we derive the relationship between the inradius of a triangle , and its three altitudes . Using an area argument, we can get the following well known result where are the side lengths of , and is the triangle's area. Substituting into the above we get Similarly, we can get Hence, \begin{align}\label{e1} \frac{1}{r}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \end{align}

Note that there exists a unique, non-degenerate triangle with altitudes if and only if are the side lengths of a non-degenerate triangle, i.e., .

With this in mind, it remains to find all positive integer solutions to the above such that , and . We do this by doing casework on the value of .

Since is a positive integer, . Since , , so . The only possible values for are 1, 2, and 3.

Case :

For this case, we can't have , since would be too small. When , we must have . When , we would have , which doesn't work. Hence this case only yields one valid solution

Case :

For this case, we can't have , for the same reason as in Case 1. When , we must have . When , we must have or . Regardless, appears, so it is not a valid solution. When , . Hence, this case also only yields one valid solution

Case :

The only possible solution is , and clearly it is a valid solution.

Hence the only valid solutions are , and our answer is

~tsun26

See also

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